Plan





PMF - Skopje
Primeri nelinearnih oscilatora
Fazni prelaz kod modela Kuramoto
Nestabilne fiksne ta~ke i wihova stabilizacija
Nau~na produkcija na Balkanu
PMF, Skopje
Prose~na golemina na
evropski oddel za fizika (2009)
Studenti - 467
(univerzitet - 23260)
Nastaven personal - 79
(univ - 1990)
Doktoranti - 75
Na PMF, soodvetno st. 20-30, n. 23 i d. 7-8
. . .
Current programme – part 1
(semesters 1-4)
(lectures + tutorials + laboratory = credit points)
I
Mechanics
Mathematical Analysis 1
Computer usage in physics
Introduction to metrology
Elective course 1
Elective course 2
III
Electromagnetism
Mathematical physics 1
Theoretical mechanics
Oscillations and waves
Elective course 6
Elective course 7
4+2+2=8
4+4+0=8
2+0+2=4
2+0+2=4
3+0+0=3
3+0+0=3
II
Molecular physics
Mathematical analysis 2
Chemistry
Elective course 3
Elective course 4
Elective course 5
4+2+2=7
3+3+0=7
3+2+0=6
2+2+0=4
3+0+0=3
3+0+0=3
IV
Optics
4+2+2=8
Mathematical physics 2
3+3+0=7
Electronics
3+1+3=7
Theoretical electrodynamics and
special theory of relativity 3+2+0=5
Elective course 8
3+0+0=3
4+2+2=8
3+3+0=7
3+0+3=6
3+0+0=3
3+0+0=3
3+0+0=3
Current programme - part 2
(semesters 5-8, physics teachers branch)
V
Atomic physics
Measurements in physics
General astronomy
Elective course 9
Elective course 10
Elective course 11
Elective course 12
4+2+2=8
3+0+3=6
2+1+0=4
3+0+0=3
3+0+0=3
3+0+0=3
3+0+0=3
VII
Use of computers in teaching
2+0+2=5
Methodology of physics teaching 1 2+2+3=8
School experiments 1
2+0+3=6
Psychology
3+2+0=5
Macedonian language
0+2+0=2
Introduction to biophysics
2+0+2=4
VI
Nuclear physics
Introduction to quantum theory
Introduction to materials
Basics of solid state physics
Pedagogy
4+2+2=8
3+2+0=6
2+0+2=5
3+1+2=6
3+2+0=5
VIII
Methodology of physics teaching 2
(school practice)
2+2+3=8
School experiments 2
2+0+3=5
Design of electronic equipment
2+0+3=4
History and philosophy of physics 3+1+0=4
Diploma thesis
0+0+9=9
Nonlinear oscillator


x  b x  sin x  A sin  t

x y

y   sin x  by  A sin  t
The Lorenz system
E. N. Lorenz, “Deterministic nonperiodic flow,”
J. Atmos. Sci. 20 (1963) 130.
Fixed points:
C0 (0,0,0)
C± (±8.485, ±8.485,27)
Eigenvalues:
l(C0) = {-22.83, 11.83, -2.67}
l(C±) = {-13.85, 0.09+10.19i, 0.09-10.19i}
Chaotic attractor of the
unperturbed system (F(t)=0)
van der Pol oscillator


x   ( x  1) x  x  0
2

x y

y   (1  x ) y  x
2
Limit cycle
Rössler oscillator with harmonic
forcing

x  y  z  E sin(ext t )

y  x  ay

z  f  z ( x  c)
Historical example from Biology
The glowworms ... Represent another shew, which settle on some
Trees, like a fiery cloud, with this surprising circumstance, that a
whole swarm of these insects, having taken possession of one
Tree, and spread themselves over its branches, sometimes hide
their Light all at once, and a moment after make it appear again
with the utmost regularity and exactness …
Engelbert Kaempfer description from his trip in Siam (1680)
Further examples
• The Moon facing the Earth;
Gallilean satelites; Kirkwood gaps
• Cyclotron and other accelerators
• Stroboscope; Fax-machine
• Biological clocks; Jet lag
• Pacemakers
• Farmacological actions of steroids
Further examples 2
• Cardiorespiratory system
• Entrainment of cardial and locomotor
rhythms
• Cardiovascular coupling during anesthesia
• Synchronization between parts of the brain
• Magnetoencephalographic fields and
muscle activity of Parkinsonian patients
Modelot na Kuramoto
Parametar na poredok i
sinhronizacija
r 1
r 0
Re{enie na modelot na Kuramoto (1975)
 /2
r  Kr 
 / 2
cos 2   g ( Kr sin  )d
re{enija
r 0
i
r0
K c  2 / g (0)
 /
g ( )  2
 r  1 Kc / K
2
 
INTRODUCTION - THE PYRAGAS CONTROL METHOD
- Time-delayed feedback control (TDFC)
- Time-delayed autosynchronization (TDAS)
K. Pyragas, Phys. Lett. A 170 (1992) 421
Applications
Delays are natural in many systems
• Coupled oscillators
• Electronic circuits
• Lasers, electrochemistry
• Networks of oscillators
• Brain and cardiac dynamics
VARIABLE DELAY FEEDBACK CONTROL OF USS
Pyragas control force:
- noninvasive for USS and periodic orbits
VDFC force:
- piezoelements, noise
- saw tooth wave:
- random wave:
- sine wave:
A. Gjurchinovski and V. Urumov – Europhys. Lett. 84, 40013 (2008)
VARIABLE DELAY FEEDBACK CONTROL OF USS
THE MECHANISM OF VDFC
DELAY MODULATIONS
THE MECHANISM OF VDFC
THE MECHANISM OF VDFC
2D UNSTABLE FOCUS WITH A DIAGONAL COUPLING
n – sufficiently large
original system :
comparison system :
Characteristic equation of the comparison system (2D focus):
THE MECHANISM OF VDFC
TDAS
VDFC
VDFC
VDFC
THE MECHANISM OF VDFC
The effect of including variable delay into TDAS for small e
• condition for the roots lying on the imaginary axis for e=0 to move to
the left half-plane as e increases from zero
CONCLUSION: the stability domain will expand in all directions within
the half-space K>K0, as soon as e is increased from zero, independent of
the precise way in which the delay is varied
THE MECHANISM OF VDFC
2D unstable focus with
l  0.1 and   
e  0 (Pyragas)
Increase of the stability
domain for small e > 0
e  0 (brown)
e  0.07 (green)
e  0.1 (yellow)
THE MECHANISM OF VDFC
eK diagrams for a saw tooth wave modulation (T0=1)
THE MECHANISM OF VDFC
THE MECHANISM OF VDFC
Stability analysis for the Lorenz system (saw tooth wave)
C0 (0,0,0)
C+ (8.485, 8.485,27)
C- (-8.485, -8.485,27)
s  10, r  28, b  8/3
THE MECHANISM OF VDFC
THE MECHANISM OF VDFC
The Rössler system
e0
e  0.5
e1
e2
(sawtooth wave)
O.E. Rössler, Phys. Lett. A 57, 397 (1976).
Fixed points:
C1 (0.007,-0.035,0.035)
C2 (5.693, -28.465,28.465)
Eigenvalues:
l(C1) = {-5.687,0.097+0.995i,0.097-0.995i}
l(C2) = {0.192,-0.00000459+5.428i, -0.00000459-5.428i}
STABILIZATION OF UPO BY VDFC
SQUARE WAVE MODULATION
• periodic change of the delay, e. g. between T0 and 2T0, K fixed (VDFC)
T(t)
t - half-period of the wave
(optimal choice: tT0)
2T0
T0
t
2t
3t
4t
t
• periodic change of the delay, K varied (VDFC + SCHUSTER, STEMMLER)
T(t)
K(t)
+
2T0
T0
t
2t
3t
4t
t
K
K/2
t
2t
3t
4t
t
STABILIZATION OF UPO BY VDFC
•PYRAGAS
F(t)=K [y(t-T0)-y(t)]
•VDFC (square wave)
F(t)=K [y(t-T(t))-y(t)]
•SCHUSTER, STEMMLER
F(t)=K(t) [y(t-T0)-y(t)]
•VDFC (square wave) + SCH-ST
F(t)=K(t) [y(t-T(t))-y(t)]
Rössler T0=5.88
STABILIZATION OF UPO BY VDFC
Rössler T0=11.75
Rössler T0=17.5
STABILIZATION OF UPO BY VDFC
K periodically varied between K and K/4 (Rössler, T0=17.5)
•VDFC + SCHUSTER
•Restricted VDFC + SCHUSTER
F(t)=K(t) Sin [y(t-T(t))-y(t)]
STABILIZATION OF UPO BY VDFC
VDFC (square wave)
t = T0
t = 2T0
t = T0/2
Rössler T0=5.88
STABILITY ANALYSIS - RDDE
Retarded delay-differential equations
•
GOAL: stabilization of unstable steady states by a variable-delay
feedback control in a nonlinear dynamical systems described by a scalar
autonomous retarded delay-differential equation (RDDE)
•
MOTIVATION: extension of the delay method to infinite dimensional
systems
•
INTEREST: frequent occurrence of scalar RDDE in numerous physical,
biological and engineering models, where the time-delays are natural
manifestation of the system’s dynamics
T. Erneux, Applied Delay Differential Equations (Springer, New York, 2009)
DELAY-DIFFERENTIAL EQUATIONS
Retarded delay-differential equations
General scalar RDDE system:
T1 ≥ 0
– constant delay time
F
– arbitrary nonlinear function of the state variable x
Linearized system around the fixed point x*:
Characteristic equation for the stability of steady state x*
of the free-running system:
A. Gjurchinovski and V. Urumov – Phys. Rev. E 81, 016209 (2010)
STABILITY ANALYSIS - RDDE
Retarded delay-differential equations
Controlled RDDE system:
u(t) – Pyragas-type feedback force with a variable time delay
K
T2
f
e
n
– feedback gain (strength of the feedback)
– nominal delay value
– periodic function with zero mean
– amplitude of the modulation
– frequency of the modulation
STABILITY ANALYSIS - RDDE
Stability of the unperturbed system
STABILITY ANALYSIS - RDDE
Stability under variable-delay feedback control
Limitation of the VDFC for RDDE systems:
•
A kind of analogue to the odd-number limitation in the case of delayed
feedback control of systems described by ordinary differential equations:
W. Just et al., Phys. Rev. Lett. 78, 203(1997)
H. Nakajima, Phys. Lett. A 232, 207 (1997)
•
… refuted recently:
B. Fiedler et al., Phys. Rev. Lett. 98, 114101 (2007).
B. Fiedler et al., Phys. Rev. E 77, 066207 (2008).
STABILITY ANALYSIS - RDDE
Representation of the control boundaries
parametrized by  = Im(l)
(K,T2) plane:
EXAMPLES AND SIMULATIONS
Mackey-Glass system
•
A model for regeneration of blood cells in patients with leukemia
M. C. Mackey and L. Glass, Science 197, 28 (1977).
•
M-G system under variable-delay feedback control:
•
For the typical values a = 0.2, b = 0.1 and c = 10, the fixed points of the freerunning system are:
•
•
•
x1 = 0 – unstable for any T1, cannot be stabilized by VDFC
x2 = +1 – stable for T1  [0,4.7082)
x3 = -1 – stable for T1  [0,4.7082)
EXAMPLES AND SIMULATIONS
Mackey-Glass system (without control)
(a) T1 = 4
(b) T1 = 8
(c) T1 = 15
(d) T1 = 23
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
T1 = 23
(a) e = 0 (TDFC)
(b) e = 0.5 (saw)
(c) e = 1 (saw)
(d) e = 2 (saw)
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
T1 = 23
(a) e = 1 (sin)
(b) e = 2 (sin)
(c) e = 1 (sqr)
(d) e = 2 (sqr)
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
K = 0.5
(a) e = 0 (TDFC)
(b) e = 2 (saw)
(c) e = 2 (sin)
(d) e = 2 (sqr)
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
saw
sin
sqr
T1 = 23, T2 = 18, K = 2, e = 2, n = 5
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
EXAMPLES AND SIMULATIONS
Mackey-Glass system (VDFC)
EXAMPLES AND SIMULATIONS
Ikeda system
•
Introduced to describe the dynamics of an optical bistable resonator,
incorporating the round-trip time of light in an optical cavity via the time
delay T1
K. Ikeda, Opt. Commun. 39, 257 (1979)
K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).
•
Ikeda system under variable-delay feedback control:
•
For  = 4 and x0 = /4, the fixed points of the free-running system are:
•
•
•
x1 = 3.05708 – stable for T1  [0, 0.82801)
x2 = 1.05136 – unstable for any T1, cannot be stabilized by VDFC
x3 = -1.86979 – stable for T1  [0, 0.54767)
EXAMPLES AND SIMULATIONS
Sprott system
•
The simplest one-parameter RDDE system with a sinusoidal nonlinearity
J. C. Sprott, Phys. Lett. A 366, 397 (2007)
•
Sprott system under variable-delay feedback control:
•
The fixed points of the free-running system are:
•
x2n = 2n – unstable for any T1, cannot be stabilized by VDFC
•
x2n+1 = (2n+1) – stable for T1  [0, /2)
FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system
Caputo fractional-order derivative:
FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system
FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional Rössler system - stability diagrams
Time-delayed feedback control
Variable delay feedback control
(sine-wave, e=1, n=10)
Desynchronisation in systems of coupled oscillators
Hindmarsh - Rose oscillators
M. Rosenblum and A. Pikovsky, Phys. Rev. Lett. 92, 114102; Phys. Rev. E 70, 041904 (2004)
Global coupling
Mean field
Delayed feedback control
Desynchronisation in systems of coupled oscillators
System of 1000
H-R oscillators
Feedback
switched on at
t=5000
Kmf=0.08
K=0.0036
t=const=72.5
Desynchronisation in systems of coupled oscillators
Time-delayed feedback control
Variable delay feedback control
(sine-wave, e=40, n=10)
X – Mean field in the absence of feedback
Xf – Mean field in the presence of feedback
Suppression coefficient
T=145 – average period of the mean field in
the absence of feedback
CONCLUSIONS AND FUTURE PROSPECTS
•
Enlarged domain for stabilization of unstable steady states in systems of
ordinary/delay/fractional differential equations in comparison with Pyragas
method and its generalizations
•
Agreement between theory and simulations for large frequencies in the
delay variability
•
The enlargement of the control domain may undergo a complex
rearrangement depending on the type of the delay modulation
•
Extended area of stabilization of periodic orbits by noninvasive variabledelay feedback control
•
Variable delay feedback control provides increased robustness in achieving
desynchronization in wider domain of parameter space in system of
coupled Hindmarsh-Rose oscillators interacting through their mean field
•
The influence of variable-delay feedback in other systems (neutral DDE,
PDE, networks, …)
•
Experimental verification
SCI publikacii od balkanski gradovi
2006-2010
vkupno
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